The Figure below show coplanar concurrent force system. Find resultant and direction.

We will use the standard convention where angles are measured counter-clockwise from the positive x-axis.

• Force 1 ($F_1 = 300$ N): The angle with the negative x-axis is $40^\circ$. So, the angle from the positive x-axis is $180^\circ - 40^\circ = 140^\circ$.

  $$F_{1x} = 300 \cos(140^\circ) = 300(-0.7660) = -229.80 \text{ N}$$ $$F_{1y} = 300 \sin(140^\circ) = 300(0.6428) = 192.84 \text{ N}$$

• Force 2 ($F_2 = 400$ N): The angle with the positive x-axis is $60^\circ$.

  $$F_{2x} = 400 \cos(60^\circ) = 400(0.5) = 200.00 \text{ N}$$ $$F_{2y} = 400 \sin(60^\circ) = 400(0.8660) = 346.40 \text{ N}$$

• Force 3 ($F_3 = 300$ N): This force acts along the positive x-axis, so its angle is $0^\circ$.

  $$F_{3x} = 300 \cos(0^\circ) = 300(1) = 300.00 \text{ N}$$ $$F_{3y} = 300 \sin(0^\circ) = 300(0) = 0 \text{ N}$$

• Force 4 ($F_4 = 200$ N): This force acts along the negative y-axis, so its angle is $270^\circ$.

  $$F_{4x} = 200 \cos(270^\circ) = 200(0) = 0 \text{ N}$$ $$F_{4y} = 200 \sin(270^\circ) = 200(-1) = -200.00 \text{ N}$$

$$\sum F_x = F_{1x} + F_{2x} + F_{3x} + F_{4x}$$ $$\sum F_x = -229.80 \text{ N} + 200.00 \text{ N} + 300.00 \text{ N} + 0 \text{ N}$$ $$\sum F_x = 270.20 \text{ N}$$

$$\sum F_y = F_{1y} + F_{2y} + F_{3y} + F_{4y}$$ $$\sum F_y = 192.84 \text{ N} + 346.40 \text{ N} + 0 \text{ N} - 200.00 \text{ N}$$ $$\sum F_y = 339.24 \text{ N}$$

$$R = \sqrt{(\sum F_x)^2 + (\sum F_y)^2}$$ $$R = \sqrt{(270.20 \text{ N})^2 + (339.24 \text{ N})^2}$$ $$R = \sqrt{73008.04 \text{ N}^2 + 115083.7776 \text{ N}^2}$$ $$R = \sqrt{188091.8176 \text{ N}^2}$$ $$R \approx 433.695 \text{ N}$$

$$\theta = \arctan\left(\frac{\sum F_y}{\sum F_x}\right)$$ $$\theta = \arctan\left(\frac{339.24 \text{ N}}{270.20 \text{ N}}\right)$$ $$\theta = \arctan(1.2555)$$ $$\theta \approx 51.46^\circ$$

Since both $\sum F_x$ and $\sum F_y$ are positive, the resultant force is in the first quadrant, and the angle is measured counter-clockwise from the positive x-axis.

The resultant force is $\boxed{\text{433.7 N}}$ at an angle of $\boxed{\text{51.46}^\circ}$ from the positive x-axis.